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Stabilité d’ondes périodiques, schéma numérique pour le chimiotactisme

Abstract : This thesis is organized around two aspects of the study of partial differentialequations. In a first part, we study the stability of periodic solutions for conservationlaws. First, we prove asymptotic L1-stability of periodic solutions of scalarinhomogeneous conservation laws. Then, we show a result on structural stability ofroll-waves. More precisely, we prove that periodic solutions of a hyperbolic systemwithout viscosity are the limits of the solutions of the problem with viscosity, as theviscous term tends to 0. In a second part, we study a system of partial differentialequations derived from biology: the model of Patlak-Keller-Segel in dimension 2, describingthe phenomena of chemotaxis. For this model, we construct a finite-volumescheme, which approaches the solution while keeping some properties of the system:positivity, conservation of mass, energy estimate.
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Valérie Le Blanc. Stabilité d’ondes périodiques, schéma numérique pour le chimiotactisme. Mathématiques générales [math.GM]. Université Claude Bernard - Lyon I, 2010. Français. ⟨NNT : 2010LYO10091⟩. ⟨tel-00845883⟩



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